Clique complexes of strongly regular graphs, their eigenvalues, and cohomology groups

It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can distinguish them. In this direction, we study the clique complexes of strongly regular graphs, and determine the spectra of the triangle complexes of several families of strongly regular graphs including Hamming graphs and Triangular graphs. In many cases, the spectrum of the triangle complex distinguishes between strongly regular graphs with the same parameters, but we find some examples where that is not the case. We also prove that if a graph has the property that for any induced cycle, there are four consecutive vertices on the cycle with a common neighbor, then the first cohomology group of the graph is trivial and we apply this result to several families of graphs.